Publisher Summary This chapter presents three mathematical tools that can be used to describe graph-grammars and the sets generated by them. The chapter describes graph properties by logical formulas and presents a comparison of the powers of several logical languages. The chapter discusses first-order logic, second-order logic, and monadic second-order logic, together with a few variants and restrictions of these three languages. It presents algebraic techniques that are useful for defining and studying graph rewriting rules and context-free graph-grammars. The chapter also presents the use of category theory for specifying graph rewriting rules in a precise and concise way, and for properly defining the least or rather the initial solution of a system of graph equations. It presents some links between the context-free graph-grammars and monadic second-order logic. Some applications to the definition of sets of graphs by forbidden configurations and to the theory of NP-completeness are also presented in the chapter.

Graph rewriting: an algebraic and logic approach

An axiomatic definition of context-free rewriting and its application to NLC graph grammars

An elementary introduction to the notion of an NLC graph grammar is given, and several of its extensions and variations are discussed in a systematic way. Simple concepts are considered rather than technical details.

https://link.springer.com/content/pdf/10.1007%2FBFb0017374.pdf

Graph Grammars Based on Node Rewriting: An Introduction to NLC Graph Grammars

An abstract notion of context-free grammar is introduced. It deals with abstract objects that can be words, trees, graphs or other combinatorial objects. It is applied to NLC graph grammars introduced by Rozenberg and Janssens. The monadic second-order theory of a context-free NLC set of graphs is decidable.

An Axiomatic Definition of Context-Free Rewriting and its Application to NLC Graph Grammars

We define a set of operations on graphs and an algebraic notation for finite graphs. A complete axiomatization of the equivalence of graph expressions by equational rules is given. Graph rewriting systems can be defined as rewriting systems on graph expressions. This new definition is equivalent to the classical one using double push-outs.

A representation of graphs by algebraic expressions and its use for graph rewriting systems

This paper investigates the basic properties of the class of neighborhood-uniform nodel label controlled (NUNLC) graph grammars The class of NUNLC grammars is distinguished by requiring a very natural restriction on the connection relations of NLC grammars The restriction implies “Church-Rosser property” of derivations in an NUNLC grammar, which makes the class of NUNLC grammars “technically easier” to investigate A number of combinatorial properties of the languages generated by the class of NUNLC grammars are proved Also, it is demonstrated that a number of basic properties are decidable for the class of NUNLC grammars—many of them are undecidable in the whole class of NLC grammars

D Janssens

Papers

Neighborhood-uniform NLC grammars